Consider the mappings I : P4 P5 and D : P P5 defined by...
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Consider the mappings I : P4 → P5 and D : P₁ → P5 defined by a1 a3 4 021²2² +2²³ +²2² + 3 4 I(p(x)) = a。x + a4 5 X 5 D(p(x)) = a₁ +2a₂x +3a3x² +4a4x³. where p(x) = a₁ + a₁x + a₂x² + a3x³ + aşxª. You may assume that both these mappings are linear. Let T : P4 → P5 be defined by T(p(x)) = I(p(x)) – ½⁄3x²D(p(x)). - (a) Prove that T is linear using the definition of linearity. Hint: You may use the fact that D and I are linear. (b) Determine a basis for the kernel of T and a basis for the range of T. Note: Make sure that you don't just find a spanning set, but you also prove it is linearly independent. Consider the mappings I : P4 → P5 and D : P₁ → P5 defined by a1 a3 4 021²2² +2²³ +²2² + 3 4 I(p(x)) = a。x + a4 5 X 5 D(p(x)) = a₁ +2a₂x +3a3x² +4a4x³. where p(x) = a₁ + a₁x + a₂x² + a3x³ + aşxª. You may assume that both these mappings are linear. Let T : P4 → P5 be defined by T(p(x)) = I(p(x)) – ½⁄3x²D(p(x)). - (a) Prove that T is linear using the definition of linearity. Hint: You may use the fact that D and I are linear. (b) Determine a basis for the kernel of T and a basis for the range of T. Note: Make sure that you don't just find a spanning set, but you also prove it is linearly independent.
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a To prove that T is linear we need to show that it satisfies the two properties of linearity additivity and homogeneity Additivity Tp1r p2r Ip1r p2r ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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